Integration of `(1)/(sqrt(x^(2)+9))` with respect to `(x^(2)+1)` is equal to

Options

A. `sqrt(x^(2)+9)+C`
B. `-(1)/(sqrt(x^(2)+9))+C`
C. `2sqrt(x^(2)+9)+C`
D. none of these

Answer

To solve the integral

\[ \int \frac{1}{\sqrt{x^2 + 9}} \, d(x^2 + 1), \]

we will follow these steps:

### Step 1: Rewrite the integral
We can express the integral in terms of \( u = x^2 + 1 \). Therefore, we have:

\[ du = 2x \, dx \quad \Rightarrow \quad \frac{du}{2} = x \, dx. \]

However, we need to express \( dx \) in terms of \( du \). We can also express \( x^2 \) in terms of \( u \):

\[ x^2 = u - 1. \]

### Step 2: Change of variable
Now, we substitute \( x^2 \) in the integral:

\[ \sqrt{x^2 + 9} = \sqrt{(u - 1) + 9} = \sqrt{u + 8}. \]

Thus, the integral becomes:

\[ \int \frac{1}{\sqrt{u + 8}} \cdot \frac{du}{2\sqrt{u - 1}}. \]

### Step 3: Simplify the integral
This integral can be simplified to:

\[ \frac{1}{2} \int \frac{1}{\sqrt{u + 8}} \, du. \]

### Step 4: Solve the integral
Using the integral formula

\[ \int x^{-1/2} \, dx = 2x^{1/2} + C, \]

we can integrate:

\[ \frac{1}{2} \cdot 2\sqrt{u + 8} + C = \sqrt{u + 8} + C. \]

### Step 5: Substitute back
Now, substitute back \( u = x^2 + 1 \):

\[ \sqrt{(x^2 + 1) + 8} + C = \sqrt{x^2 + 9} + C. \]

### Final Answer
Thus, the integral

\[ \int \frac{1}{\sqrt{x^2 + 9}} \, d(x^2 + 1) = \sqrt{x^2 + 9} + C. \]